Use of Higher Overvoltages in Coulostatic Kinetic Measurements J. M. Kudirka and C. G. Enke Department of Chemistry, Michigan State Uniuersity, East Lansing, Mich. 48823
FORA FIRST ORDER electrode reaction, the rate of electron transfer is given by the following well known equation ( I ) : i f
=
35
nFAksCO*(l-a)C~*a{ (Co/Co*)exp(-crnFq/RT) -
30
(CR/CR*)exP[(l - a)nFqIRTl] (1) where if is the faradaic current; a , the transfer coefficient; k,, the standard electrochemical rate constant ; q , overpotential; CRand Co,surface concentration of reduced and oxidized species; CR* and Co*, bulk concentrations of reduced and oxidized species; and the other terms have their usual meaning. When diffusion also must be considered, this equation constitutes a boundary condition for solution of the Fick diffusion equations. Because the relationship between potential and concentration in Equation l is nonlinear, in general the Fick boundary value problem cannot be solved analytically. Thus in deriving the basic equation describing simultaneous charge transfer and diffusional relaxation in the coulostatic method, it is common practice to linearize Equation l by expanding the exponentials in a Taylor series and retaining only the first two terms ( 2 , 3). For this approximation t o be valid the argument of the exponentials must be much less than 1. This has limited all perturbation techniques using the linearized equation to small potential excursions of about 5/n mV. By solving the boundary value problem without introducing linearization, Nicholson ( 4 ) has found that for the case of pure diffusional relaxation, the exponential terms are of minor importance in the final result so that larger overpotentials can be used without increasing the error level. The results agree with linearized theory for overpotentials as large as 25jn mV. To determine if the same is true of the case for simultaneous charge transfer and diffusional relaxation, it was decided to solve the general equation for this case without introducing linearization. The boundary value problem to be solved is the following: dCo/bt
=
Do(b2Co/dx2)
(2)
bCR/bf
=
DR(~~CR/~X')
(3)
t = 0, x t
t
> 0, C O
> 0, x *
> 0, x
=
CO
20
> E
6
15
10 5
2
4
8
6
10
12
16
14
t , p sec.
Figure 1. Comparison of linear coulostatic decay curve with experimental data for the hexacyanoferrate (III/II) couple, CO = C R = 1 X 10-5 m/cma, Cd = 28.6 pF/cmZ, I" = 0.235 A/cm2,experimental, - calculated from Equation 20 species, and S(8,r) is the Dirac delta function. The integral equation representations of Equations 2 and 3 are (5):
.-
rt
where r is a dummy variable of integration. also be defined as:
The flux can
D o ( X O / ~ X=) kfCO(O, t ) - kbCR(0, t )
(11)
where kf
=
ksexp[(-anF/RT) (E - E")]
(12)
and kb = k,exp[(l - a ) (nF/RT) ( E - E")]
(13)
(4)
* c O * , CR * c R *
(5)
By combining Equations 7, 8, 9, 10, 11, 12, and 13 and rearranging, the following equation is obtained :
(6)
TC(dV/dt) - 7 c V t = O W , t> =
cO*,
0, Do(bCo/bx)
COICR = exp[(nF/RT) (E
t
cR*
=
CR
25
I
=
+ D R ( ~ C R / ~=X0)
- E")] =
exp[(-crnF/RT)q] RT/nF - TO^^^/^^/^
( C O * / C R * ) ~ ~ P [ ~ F ~ /(7) RTI Do(bCo/d~)= (Cd/nFA)dq/dt - (Cd/nFA)qt= OS@, 1)
(8)
The following parameters have not been defined previously : Cd is the double layer capacity, q t = o is the maximum potential developed across the double layer before relaxation occurs, Do and DR the diffusion coefficients for oxidized and reduced
[l'
{
dq/dt d r / d G
- vt=o
lts(e,
t ) d T / d ~]
RTn/F exp[(nF/RT)q] - exp[(nFiRT)q1TR1'2ia1'zx
(14) (1) P. Delahay, "New Instrumental Methods in Electrochemistry," Interscience Publishers, Inc. New York, N.Y., 1954, p 35. (2) P. Delahay, J . Phys. Chem., 66, 2204 (3962). (3) W. H. Reinmuth, ANAL.CHEM.,34, 1272 (1962). (4) R. S.Nicholson, ibid., 37, 667 (1965). 614
0
ANALYTICAL CHEMISTRY, VOL. 44, NO. 3, MARCH 1972
where rc = RTCd/nFi, ( 5 ) W. H. Reinmuth, ibid., 34, 1446 (1962).
(15)
Table I. Comparison of Linearized and Nonlinearized Coulostatic Decay Curves as n/nt 50
Te/Td
f/7d
0.01 0.04 0.16 0.36 0.64 1.oo 2.25 4.00 6.25 9.00
io
Nonlinear Eq 19
Linear Eq 20
Nonlinear Eq 19
Linear Eq 20
Nonlinear Eq 19
Linear Eq 20
Nonlinear Eq 19
Linear Eq 20
Nonlinear Eq 19
0.999 0.999 0.997 0.993 0.987 0.980 0.957 0.925 0.887 0.842
0.999 0.999 0.997 0.993 0.987 0.980 0.958 0.926 0.889 0.845
0.998 0.992 0.970 0,936 0.892 0.841 0.698 0.555 0.430 0.332
0.998 0.992 0.970 0.939 0.894 0.842 0.699 0.557 0.432 0.335
0.991 0.966 0.886 0.786 0.686 0.593 0.417 0.309 0.242 0.198
0.991 0.967 0.888 0,787 0.689 0.595 0.420 0.313 0.247 0.202
0.982 0.941 0.825 0.705 0.599 0.513 0.365 0.279 0.225 0.188
0.982 0.942 0.827 0.707 0.601 0.514 0.369 0.283 0.227 0.190
0.927 0.838 0.692 0.583 0.499 0.435 0.326 0.256 0.205 0.168
0.929 0.840 0.695 0.586 0.502 0.437 0.328 0.258 0.207 0.169
- E")]
(16)
Et=E-v
roil2
=
R T Cd/n2F2A(l/CoDo1i2)
(17)
T
=~ RT~ cd/n2F2A(1/CRDRli2) ~
(18)
R
Now substituting $(t) = (nF/RT)v and $t = (nF/RT)vt=o into Equation 14, replacing f with A, and integrating from zero to t, a nonlinear integral equation is obtained:
By reducing this to dimensionless form and using the method of numerical solution (4), curves can be obtained which follow this equation. The linear equation derived by Delahay (2) and Reinmuth (3): 7 = vt=o(y
0.05
0.5
Linear Eq 20
nFA Co*ksexp[(-crnF/RT) (Ei
=
1
5
0
- P)-I[y exp(P2t)erfc@r1/2)P exp(y2t)erfc(yt'/*)I
(20)
+
Table I for various rations of rCto Td, This ratio has been shown to be a very useful indicator of the rate controlling process and of the most efficient data analysis method (6). The examples given in Table I cover the range from almost complete charge transfer control (Tc/Td) = 50) to almost complete diffusion control ( T c / T d = 0.05). For n = 1 and v t = 0 25.6 mV, the linear and nonlinear relaxation curves agree within 2 %; thus it appears that higher overpotentials can be used in the coulostatic method where either or both charge transfer and diffusional relaxation are occurring without introducing error. To test the use of the linearized equation for large overpotentials, the hexacyanoferrate (III)J(II)couple on platinum was studied. Experimental conditions were as given by Daum and Enke (7). Overpotentials from 4 to 25 mV were used and the capacitance and exchange current calculated using a computer curve fitting program for Equation 20. Figure 1 indicates the kind of results which were obtained at higher overpotentials. This result is essentially the same as those obtained using low overpotentials. Thus overpotentials as high as 25/n mV can be used in the coulostatic method and accurate results obtained. ACKNOWLEDGMENT
The authors thank R. S. Nicholson for his assistance in suggesting methods for the numerical solution of the nonlinear integral equation.
TR112
RECEIVED for review June 4, 1971. Accepted September 24, 1971.
can also be solved to give similar curves. Typical results from computer solution of the nonlinear integral Equation 19 and Equation 20 of Delahay and Reinmuth are given in
(6) J. M. Kudirka, P. H. Daum, and C. G. Enke, ANAL.CHEM., 44, 309 (1972). (7) P. H. Daum and C. G. Enke, ibid.,41, 653 (1969).
where/3,y
= Td112/2Tc =t
and f d l / '
=
roil2
ANALYTICAL CHEMISTRY, VOL. 44, NO. 3, MARCH 1972
615
